An abelian way approach to study random extended intervals and their ARMA processes

Abstract

An extended interval is a range $ A = [\underline{A}, \overline{A}] $ where $ \underline{A} $ may be bigger than $ \overline{A} $. This is not really natural, but is what has been used as the definition of an extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product $ {\mathbb R}\times {\mathbb Z}_2 $, where $ {\mathbb Z}_2 = \{0, 1\} $ is the set of directions; the direction $ 0 $ is for increasing intervals, and the direction $ 1 $ for decreasing ones. For instance, $ [3, 6]\times\{1\} $ is the decreasing version of $ [6, 3] $. Thereafter, we introduce on the set of extended intervals a family of metrics $ d_\gamma $, depending on a function $ \gamma(t) $, and show that there exists a unique metric $ d_\gamma $ for which $ \gamma(t)dt $ is what we have called an "adapted measure". This unique metric has very good properties, is simple to compute, and has been implemented in the software $ R $. Furthermore, we use this metric to {define variability for random extended intervals. We further study extended interval-valued ARMA} time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.

Keywords

• random set
• random extended interval
• distance
• measure
• time series